The vanishing levels of a tree

Abstract

We initiate the study of the spectrum $Vspec(\kappa)$ of sets that can be realized as the vanishing levels $V(T)$ of a normal $\kappa$-tree $T$. The latter is an invariant in the sense that if $T$ and $T'$ are club-isomorphic, then the symmetric difference of $V(T)$ and $V(T')$ is nonstationary. Additional features of this invariant imply that $Vspec(\kappa)$ is closed under finite unions and intersections. The set $V(T)$ must be stationary for an homogeneous normal $\kappa$-Aronszajn tree $T$, and if there exists a special $\kappa$-Aronszajn tree, then there exists one $T$ that is homogeneous and satisfies $V(T)=\kappa$ (modulo clubs). It is consistent (from large cardinals) that there is an $\aleph_2$-Souslin tree, and yet $V(T)$ is co-stationary for every $\aleph_2$-tree $\mathbf T$. Both $V(T)=\emptyset$ and $V(T)=\kappa$ (modulo clubs) are shown to be feasible using $\kappa$-Souslin trees even at some large cardinal close to a weakly compact. It is also possible to have a family of $2^\kappa$ many $\kappa$-Souslin trees for which the corresponding family of vanishing levels forms an antichain modulo clubs.